Solve the following equations: + =1 - Vidyadip

Question

Solve the following equations:
$\sin\text{x}+\cos\text{x}=1$

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Answer

We have,
$\sin\text{x}+\cos\text{x}=1$
divide both side by $\sqrt{2},$ we get,
$\Rightarrow\frac{1}{\sqrt{2}}\sin\text{x}+\frac{1}{\sqrt{2}}\cos\text{x}=\frac{1}{\sqrt{2}}$
$\Rightarrow\sin\frac{\pi}{4}\sin\text{x}+\cos\frac{\pi}{4}\cos\text{x}=\frac{1}{\sqrt{2}}$
$\Rightarrow\cos\Big(\text{x}-\frac{\pi}{4}\Big)=\cos\frac{\pi}{4}$
$\Rightarrow\text{x}=\frac{\pi}{4}=2\text{n}\pi\pm\frac{\pi}{4},\text{n}\in\text{z}$
$\Rightarrow\text{x}=2\text{n}\pi+\frac{\pi}{2}$ or $2\text{n}\pi,\text{n}\in\text{z}$

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